A gas tagging failed fuel assembly detection system 10 is shown in simplified conceptual form in FIG. 1. Gas tagging for failed fuel identification in nuclear reactors involves the addition of a small amount of stable noble gas isotopes to the helium fill gas of each of the reactor's fuel rods 12. The same composition of tag gas isotopes is used in every rod 12 in a given fuel assembly 14, with the gas composition systematically varied from one assembly to the next throughout the nuclear reactor 16. When a fuel rod 12 fails, it releases a portion of its tag gas 11 into the reactor's primary coolant system 18. Using a mass spectrometer 20, the detection system 10 "sniffs" a small gas sample from the primary coolant system to determine the assembly (or assemblies) containing the leaking fuel rod(s).
Each gas tag has a specified isotopic composition in terms of the ratios of the constituent isotopes with respect to one another, or, equivalently, in terms of the mole percent of each constituent isotope. To produce a canister of tag gas with the desired isotopic composition, several canisters of feed gas from a commercial enriched-gas supplier are used. Each feed gas blend has a unique isotopic composition determined by the physical enrichment process (either thermal diffusion or gas centrifuge) used to enhance the abundance of the individual isotopes for each noble gas species. In general, there is a very wide range of costs for each available feed gas blend. The unit costs of feed gas blends can easily vary by three orders of magnitude between relatively low cost natural-abundance feed gases (whose mole percents are essentially the same as those which occur naturally in the air, requiring no enrichment) and more expensive highly enriched blends, which have high mole fractions of those isotopes characterized by low natural abundance.
For purposes of illustration, Table I shows a set of eight hypothetical feed gas mixtures, each of which comprises four isotopes. The isotopic compositions for each feed gas are typical of those obtainable from an enriched-gas supplier by thermal diffusion, and the range of costs is typical of the costs for enriched stable noble gases. While Table I contains information relating to the gas blending procedure for eight hypothetical four-component feed gases, the method discussed herein is applicable to virtually any number of feed gas mixtures comprising any number of constituent components.
TABLE I ______________________________________ Costs and Enrichments for Eight Sample Feed Gas Mixtures Feed Gas Mole % Cost Mixture No. Isotope 1 Isotope 2 Isotope 3 Isotope 4 $/L ______________________________________ 1 10 35 15 40 $800 2 20 30 12 38 1350 3 30 27 11 32 2100 4 40 24 16 20 3400 5 50 20 20 10 5500 6 60 15 15 10 6800 7 70 12 8 10 9500 8 80 6 6 8 11200 ______________________________________
There are a fixed number of noble gas isotopic mixtures commercially available which encompass a wide range of costs. To obtain a target composition for a desired gas tag, there are infinitely many ways to blend the eight available feed gases and produce the required isotopic mole percents. These blends span a very large range of costs. A non-trivial optimization problem arises when the objective is to find which combination of eight or less input gases produces the minimum overall tag cost, while satisfying the analytical constraints imposed by the target isotopic ratios and the physical constraint that the proportion of each feed gas must be non-negative.
The prior art in this area employs two approaches for solving the tag-blend optimization problem, each of which has limitations when employed in a commercial production-scale operation. One approach, developed at the Experimental Breeder Reactor-II (EBR-II) at Argonne National Laboratory in the early 1970's, involves computing a graphical control chart capable of accommodating only three feed gases and minimizing the use of the most expensive isotope. An example of this control chart for EBR-II's xenon gas tags is shown in graphic form in FIG. 2. This control-chart approach was relatively convenient at the time it was devised before the advent of personal computers, but produced tag-blends that were suboptimal with respect to cost. The cost penalties estimated by comparing control-chart blends with blends obtained from optimization techniques described below, averaged 30% of the tag cost when the same three fuel gas blends are used. This is not a severe penalty for a small research reactor such as EBR-II, but results in sizable economic penalties for a commercial type reactor involving the use of as many as 10.sup.5 pins per core where the cost difference per pin may be as great as $2.00.
A second prior art attempt at tag-blend optimization involves a computer implementation of a detailed analytical solution to the constrained optimization problem which suffers from computational complexity. In this approach, a problem involving ten feed gases with five analytical constraints is mapped onto a system of 15 nonlinear simultaneous equations, which is transformed into a system of 15 linear simultaneous ordinary differential equations (ODEs). One difficulty in this approach is that the solution requires an input "guess" of the solution vector. For constrained optimization problems of this nature, there are very many widely separated local minima and there is no way, a priori, to select a starting vector that will ensure convergence to the overall global minimum. To avoid this difficulty prior approaches sought to supply a very large number of starting-guess vectors and solve the complete system of 15 ODEs for all starting vectors. This approach, which requires a large supercomputer to blend a small system of tags, would not be practical for a large scale commercial operation.
Still another problem with prior art approaches to tag gas manufacture involves the actual physical procedure employed in blending the tags after a blend composition is determined by one of the procedures discussed above. The blending process entails manual connection and disconnection of gas cylinders, and opening and closing of a complex system of valves connecting purge lines, feed-gas lines, vacuum lines and compressor lines. This procedure is lengthy, manpower intensive, error prone, wasteful of gas, and laborious for the gas chemist.
The present invention addresses the aforementioned limitations of the prior art by determining the most cost-effective isotopic mixtures for nuclear reactor gas tags for computer controlling the mixing of the isotopes from a number of filling gas canisters to a single blended tag canister. The latter is used to fill a fuel pin with a tag of that composition, thereby providing it with a unique identity to permit detection of a leak from a given pin and allow for identification of the leaking pin, or pins.